Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Title: Permutations and Divisibility**
**Objective:**
Prove that if \( r \leq s \leq n \), then \( P(n, s) \) is divisible by \( P(n, r) \).
----
**Explanation:**
The problem involves proving a property related to permutations. Here, \( P(n, s) \) represents the number of permutations of \( n \) items taken \( s \) at a time. Similarly, \( P(n, r) \) represents permutations of \( n \) items taken \( r \) at a time.
**Steps to Approach the Proof:**
1. **Understand Permutations:**
- The formula for permutations \( P(n, k) \) is given by:
\[
P(n, k) = \frac{n!}{(n-k)!}
\]
- This formula represents selecting \( k \) items from \( n \) without replacement and considering the order.
2. **Divisibility Condition:**
- To prove that \( P(n, s) \) is divisible by \( P(n, r) \), show that:
\[
\frac{P(n, s)}{P(n, r)}
\]
is an integer.
3. **Simplifying the Expressions:**
- Write the expressions for \( P(n, s) \) and \( P(n, r) \):
\[
P(n, s) = \frac{n!}{(n-s)!}
\]
\[
P(n, r) = \frac{n!}{(n-r)!}
\]
- Compute the ratio:
\[
\frac{P(n, s)}{P(n, r)} = \frac{(n-r)!}{(n-s)!}
\]
4. **Prove the Ratio is Integer:**
- Note that since \( r \leq s \), each step from \( (n-r)! \) to \( (n-s)! \) is a product of consecutive integers:
\[
(n-s+1)(n-s+2)\cdots(n-r)
\]
- The product is clearly an integer, confirming divisibility.
This completes the proof that \( P(n, s) \) is divisible by \( P(n, r) \) under the given conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb48054c1-c883-40e2-8364-5c805e917eb2%2F66b02d4b-a13f-424b-978e-2f2a8c84a7af%2Fqvub9al_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Permutations and Divisibility**
**Objective:**
Prove that if \( r \leq s \leq n \), then \( P(n, s) \) is divisible by \( P(n, r) \).
----
**Explanation:**
The problem involves proving a property related to permutations. Here, \( P(n, s) \) represents the number of permutations of \( n \) items taken \( s \) at a time. Similarly, \( P(n, r) \) represents permutations of \( n \) items taken \( r \) at a time.
**Steps to Approach the Proof:**
1. **Understand Permutations:**
- The formula for permutations \( P(n, k) \) is given by:
\[
P(n, k) = \frac{n!}{(n-k)!}
\]
- This formula represents selecting \( k \) items from \( n \) without replacement and considering the order.
2. **Divisibility Condition:**
- To prove that \( P(n, s) \) is divisible by \( P(n, r) \), show that:
\[
\frac{P(n, s)}{P(n, r)}
\]
is an integer.
3. **Simplifying the Expressions:**
- Write the expressions for \( P(n, s) \) and \( P(n, r) \):
\[
P(n, s) = \frac{n!}{(n-s)!}
\]
\[
P(n, r) = \frac{n!}{(n-r)!}
\]
- Compute the ratio:
\[
\frac{P(n, s)}{P(n, r)} = \frac{(n-r)!}{(n-s)!}
\]
4. **Prove the Ratio is Integer:**
- Note that since \( r \leq s \), each step from \( (n-r)! \) to \( (n-s)! \) is a product of consecutive integers:
\[
(n-s+1)(n-s+2)\cdots(n-r)
\]
- The product is clearly an integer, confirming divisibility.
This completes the proof that \( P(n, s) \) is divisible by \( P(n, r) \) under the given conditions.
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